经典多面体与量子多面体
Classical and Quantum Polyhedra
摘要：
由角动量算符构成的量子多面体是圈量子引力所倡导的量子描述中空间的构建块。在这里，我们推广了以前关于量子多面体的半经典性质的结果。关于四面体，我们比较的结果从一个典型的量子化的经典系统与最近的波函数为基础的方法的大体积部门的量子系统。这两种方法同意在导致有效运营商（由谐振子）的领导秩序，而较小的差异发生在更高的修正。微扰包含这样的修正，提高了近似的本征态。此外，这两种方法的比较也导致一个完整的波函数描述的本征态的（平方）体积运营商在负本征值的大模。对于用离散角动量量子数描述的一般量子多面体的情形，我们构造了一组满足半经典体系中动量和位置之间标准对易关系的量子算符.与以前的公式不同，这里的位置变量被选择为具有（普朗克）长度平方的维度，这有助于识别量子校正。最后，我们提供了五面体体积的Kapovich-Millson变量的表达式。
英文摘要：
Quantum polyhedra constructed from angular momentum operators are the building blocks of space in its quantum description as advocated by Loop Quantum Gravity. Here we extend previous results on the semiclassical properties of quantum polyhedra. Regarding tetrahedra, we compare the results from a canonical quantization of the classical system with a recent wave function based approach to the large-volume sector of the quantum system. Both methods agree in the leading order of the resulting effective operator (given by an harmonic oscillator), while minor differences occur in higher corrections. Perturbative inclusion of such corrections improves the approximation to the eigenstates. Moreover, the comparison of both methods leads also to a full wave function description of the eigenstates of the (square of the) volume operator at negative eigenvalues of large modulus. For the case of general quantum polyhedra described by discrete angular momentum quantum numbers we formulate a set of quantum operators fulfilling in the semiclassical regime the standard commutation relations between momentum and position. Differently from previous formulations, the position variable here is chosen to have dimension of (Planck) length squared which facilitates the identification of quantum corrections. Finally, we provide expressions for the pentahedral volume in terms of Kapovich-Millson variables.
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